Average Error: 53.1 → 0.2
Time: 6.4s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.01182013745483679:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} - \frac{-0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 0.0010145911176515101:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{x + \mathsf{hypot}\left(x, \sqrt{1}\right)}\right) + \log \left(\sqrt{x + \mathsf{hypot}\left(x, \sqrt{1}\right)}\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.01182013745483679:\\
\;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} - \frac{-0.0625}{{x}^{5}}\right)\right)\\

\mathbf{elif}\;x \le 0.0010145911176515101:\\
\;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\

\mathbf{else}:\\
\;\;\;\;\log \left(\sqrt{x + \mathsf{hypot}\left(x, \sqrt{1}\right)}\right) + \log \left(\sqrt{x + \mathsf{hypot}\left(x, \sqrt{1}\right)}\right)\\

\end{array}
double f(double x) {
        double r196941 = x;
        double r196942 = r196941 * r196941;
        double r196943 = 1.0;
        double r196944 = r196942 + r196943;
        double r196945 = sqrt(r196944);
        double r196946 = r196941 + r196945;
        double r196947 = log(r196946);
        return r196947;
}
↓
double f(double x) {
        double r196948 = x;
        double r196949 = -1.0118201374548368;
        bool r196950 = r196948 <= r196949;
        double r196951 = 0.125;
        double r196952 = 3.0;
        double r196953 = pow(r196948, r196952);
        double r196954 = r196951 / r196953;
        double r196955 = 0.5;
        double r196956 = r196955 / r196948;
        double r196957 = 0.0625;
        double r196958 = -r196957;
        double r196959 = 5.0;
        double r196960 = pow(r196948, r196959);
        double r196961 = r196958 / r196960;
        double r196962 = r196956 - r196961;
        double r196963 = r196954 - r196962;
        double r196964 = log(r196963);
        double r196965 = 0.00101459111765151;
        bool r196966 = r196948 <= r196965;
        double r196967 = 1.0;
        double r196968 = sqrt(r196967);
        double r196969 = log(r196968);
        double r196970 = r196948 / r196968;
        double r196971 = r196969 + r196970;
        double r196972 = 0.16666666666666666;
        double r196973 = pow(r196968, r196952);
        double r196974 = r196953 / r196973;
        double r196975 = r196972 * r196974;
        double r196976 = r196971 - r196975;
        double r196977 = hypot(r196948, r196968);
        double r196978 = r196948 + r196977;
        double r196979 = sqrt(r196978);
        double r196980 = log(r196979);
        double r196981 = r196980 + r196980;
        double r196982 = r196966 ? r196976 : r196981;
        double r196983 = r196950 ? r196964 : r196982;
        return r196983;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original53.1
Target45.4
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;x \lt 0.0:\\ \;\;\;\;\log \left(\frac{-1}{x - \sqrt{x \cdot x + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + 1}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.0118201374548368

    1. Initial program 63.0

      \[\log \left(x + \sqrt{x \cdot x + 1}\right)\]

    2. Taylor expanded around -inf 0.3

      \[\leadsto \log \color{blue}{\left(0.125 \cdot \frac{1}{{x}^{3}} - \left(0.5 \cdot \frac{1}{x} + 0.0625 \cdot \frac{1}{{x}^{5}}\right)\right)}\]

    3. Simplified0.3

      \[\leadsto \log \color{blue}{\left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} - \frac{-0.0625}{{x}^{5}}\right)\right)}\]

    if -1.0118201374548368 < x < 0.00101459111765151

    1. Initial program 58.9

      \[\log \left(x + \sqrt{x \cdot x + 1}\right)\]

    2. Taylor expanded around 0 0.2

      \[\leadsto \color{blue}{\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}}\]

    if 0.00101459111765151 < x

    1. Initial program 31.8

      \[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt31.8

      \[\leadsto \log \left(x + \sqrt{x \cdot x + \color{blue}{\sqrt{1} \cdot \sqrt{1}}}\right)\]

    4. Applied hypot-def0.1

      \[\leadsto \log \left(x + \color{blue}{\mathsf{hypot}\left(x, \sqrt{1}\right)}\right)\]
    5. Using strategy rm

    6. Applied add-sqr-sqrt0.1

      \[\leadsto \log \color{blue}{\left(\sqrt{x + \mathsf{hypot}\left(x, \sqrt{1}\right)} \cdot \sqrt{x + \mathsf{hypot}\left(x, \sqrt{1}\right)}\right)}\]

    7. Applied log-prod0.1

      \[\leadsto \color{blue}{\log \left(\sqrt{x + \mathsf{hypot}\left(x, \sqrt{1}\right)}\right) + \log \left(\sqrt{x + \mathsf{hypot}\left(x, \sqrt{1}\right)}\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.01182013745483679:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} - \frac{-0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 0.0010145911176515101:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{x + \mathsf{hypot}\left(x, \sqrt{1}\right)}\right) + \log \left(\sqrt{x + \mathsf{hypot}\left(x, \sqrt{1}\right)}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020036 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arcsine"
  :precision binary64

  :herbie-target
  (if (< x 0.0) (log (/ -1 (- x (sqrt (+ (* x x) 1))))) (log (+ x (sqrt (+ (* x x) 1)))))

  (log (+ x (sqrt (+ (* x x) 1)))))